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REVIEW 6 major objections 5 minor 7 references

Some mathematical issues regarding a new approach towards quantum foundations

T0 review · 6 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Two complementary maximal accessible variables, together with weak symmetry and invariant-measure conditions, force every accessible variable to be represented by a unique symmetric operator on a Hilbert space.

desk verdict The central derivation breaks on an unjustified inversion step, but the paper is a clean statement of Helland's weakened postulates and a fair target for a serious referee. read the letter →

arxiv 2411.13113 v4 pith:KEWTBANB submitted 2024-11-20 quant-ph

Inge S. Helland This is my paper · ORCID
classification quant-ph MSC 81P0581P1081P15 PACS 03.65.Ta03.65.-w
keywords accessibletheoreticalvariablescomplementaryHilbertspaceformalismquantumfoundationsBornruleepistemicinterpretationBellexperimentsymmetricoperators
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper tries to establish that the Hilbert space formalism of quantum mechanics follows from very weak assumptions about what an observer can measure. Its primitive notion is a theoretical variable, a quantity attached to an observer or a communicating group; a variable is accessible when it can in principle be measured with arbitrary accuracy. The central claim is that the existence of two complementary maximal accessible variables, together with mild group-action and invariant-measure conditions, forces a Hilbert space to exist and assigns every accessible variable a unique symmetric operator. If the argument is correct, the quantum formalism is not an axiom about microscopic systems but a consequence of the relation between complementary measurement questions. The same mathematics also yields the Born rule under two additional probabilistic postulates and supports a broadly epistemic interpretation of quantum theory.

What carries the argument

The load-bearing object is the relation of being related between two maximal accessible variables: $\theta = f(\varphi)$ and $\eta = f(k\varphi)$ for a single function $f$ on $\Omega_\varphi$ and a transformation $k$ acting there. Proposition 1 is the bridge that gets from the equal-category condition to a related pair, up to a bijective change of one variable. Once a related pair exists, the earlier representation theorem supplies the Hilbert space, and the paper constructs its required unitary representation explicitly as $U(g)f(\theta)=f(g^{-1}\theta)$ on $L^2(\Omega_\theta,\mu)$; the coherent states $U(g)f_0$ are in one-to-one correspondence with the group elements, which is exactly the condition the representation theorem needs. From that point the symmetric operator $A_\zeta$ for every accessible variable $\zeta$ is defined through the spectral apparatus.

What would settle it

Construct a finite instance satisfying the postulates in which $\Omega_\varphi$ is larger than $\Omega_\theta$, $f$ is many-to-one, and for no transformation $k$ in the acting group $M$ does $\eta(\varphi) = f(k\varphi)$ hold. If such an instance exists, Proposition 1 cannot deliver a related pair, and Theorem 1's route to a Hilbert-space representation collapses.

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Extended reading notes

Core claim

The paper's central claim is Theorem 1: assume an inaccessible background variable $\varphi$ exists, all accessible variables are functions of it, and a group $M$ acts on its range space $\Omega_\varphi$; let $\theta$ and $\eta$ be two maximal accessible variables whose range spaces have the same category and which are not in one-to-one correspondence; let $\theta$ carry a transitive group $G$ with trivial isotropy and a left-invariant measure $\mu$. Then there is a Hilbert space $H$, and every accessible variable $\zeta$ has a unique symmetric operator $A_\zeta$ on $H$. The proof converts the same-category assumption, through Proposition 1, into the statement that $\theta$ and $\eta$ are related: each is the image of $\varphi$ under one function $f$, with one image shifted by a transformation $k$ on $\Omega_\varphi$. The needed unitary representation of $G$ is then constructed explicitly on $L^2(\Omega_\theta,\mu)$ by left translation, so the theorem's conclusion follows from the earlier related-variable representation theorem. The paper states that this is the weakest possible theorem providing a foundation for the Hilbert space formalism: no pre-existing representation of $G$, no full superposition principle, and no microscopic dynamical assumptions are required.

Load-bearing premise

The load-bearing premise is that two maximal accessible variables whose range spaces have the same category can always be converted into a related pair by one function and one transformation of the hidden variable; this step needs an inverse function that the stated assumptions do not guarantee.

Editorial extensions

If this is right

  • Every accessible variable in a context with two complementary maximal variables receives a unique symmetric operator, so observables like position, momentum, and spin components arise from the structure of measurement questions rather than from a pre-assumed operator algebra.
  • Complementary maximal variables that are not bijective functions of each other produce noncommuting operators, recovering the standard noncommutativity of quantum observables.
  • Adding the generalized likelihood principle and a Dutch Book rationality condition yields the Born rule, with transition probabilities given by squared absolute values of inner products.
  • Related variables are connected by unitary similarity transformations, and the inverse statement holds for finite-dimensional variables, so changing from one maximal perspective to another is represented by a unitary change of basis.
  • The observer-limitation corollary gives an explanation of the Bell experiment's CHSH violation without invoking nonlocality: no observer can simultaneously hold two related but not mutually related maximal accessible variables.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the theorem is right, the same postulates should reproduce quantum-like structure in any field with complementary maximal variables, so decision theory, biology, and cognition become testable arenas for the formalism.
  • The most fragile point in the derivation is Proposition 1's inverse step; a counterexample with non-injective $f$ would force the theorem to require an additional injectivity or orbit assumption and would narrow the 'weakest possible' claim.
  • The paper pairs variables but does not construct the Hilbert space for three or more mutually complementary variables; an extension worth testing is whether triple complementarity forces finite-dimensional spin-like constraints and specific operator relations.
  • Theorem 7 has a behavioral prediction not drawn in the paper: at a fixed moment a person should be unable to simultaneously entertain two related but incompatible decision frames, which could be probed in decision experiments.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

6 major / 5 minor

Summary. Drawing on a series of earlier papers by the same author, this manuscript attempts to reconstruct the Hilbert-space formalism of quantum mechanics from a small set of postulates about "theoretical variables." Postulate 1 assumes an inaccessible variable φ and a group M on Ωφ; Postulate 3 gives each maximal accessible variable θ a transitive group G with trivial isotropy and a left-invariant measure; Postulate 4 requires the range spaces of two complementary maximal variables θ and η to have the same category. Proposition 1 claims that such variables can be reduced to a "related" pair θ=f(φ), ξ=f(kφ), and Theorem 1 then asserts the existence of a Hilbert space H and a unique symmetric operator for every accessible variable. Section 3 adds a generalized likelihood principle and a "superior actor" postulate to derive the Born rule; Section 5 applies the framework to the Bell experiment and to decision theory. The paper's central mathematical claim is Proposition 1/Theorem 1, and the remainder builds on it.

Significance. The project is significant in ambition: a derivation of the Hilbert-space apparatus from weak, explicitly stated postulates would be a valuable contribution to quantum foundations, and the paper is transparent about its axioms and about the limitation that the superposition principle is not fully assumed. It also gives explicit attention to the distinction between symmetric and self-adjoint operators. However, the central proof is not self-contained and contains a load-bearing gap in Proposition 1; the Hilbert space used is not fixed; and several theorems are either proved only in special cases or deferred to external papers. These are not merely presentational issues, because Theorem 1 is the foundation for the rest of the paper. The claimed "weakest possible theorem" status is not established by any minimality argument.

major comments (6)
  1. [Section 2, Proposition 1] The proof chooses a function f with θ=f(φ) and then states that for each φ1 there is a φ2 with η(φ1)=f(φ2), "since {η(φ)} has the same category as {θ(φ)}={f(φ)}." Same category, even understood as equinumerosity of the two ranges, does not imply that the range of η is contained in the range of f; two sets of the same cardinality may be disjoint. The existence of φ2 is therefore not established. This step is load-bearing because the rest of Proposition 1, and hence the reduction to a "related" pair that allows Theorem 1 to invoke Theorem 4 of Helland (2024a), depends on it.
  2. [Section 2, Proposition 1] The proof uses f^{-1} in the expression η(φ1)=f(a(f^{-1}(ξ(φ1)))) and justifies it by saying that the range of ξ has the same category as the range of η. No injectivity of f is established, and equality of cardinalities (or categorical equivalence) does not make a non-injective function invertible. In the different-orbit branch, the symbol a is first introduced as a function characterising orbits and is then treated as a point φ2=a(φ3), conflating a function with a value. Both steps are needed for the claimed reduction of θ and η to a related pair, so Proposition 1 is not proved.
  3. [Theorem 1] The proof is not self-contained. It says "Then it follows from Theorem 4 of Helland (2024a)" and later "Theorem 1 now follows from Theorem 4 of Helland (2024a) and Proposition 2," but the hypotheses, statement, and proof of that external theorem are not included. In addition, the Hilbert space is not consistently specified: Proposition 2 constructs the representation U on L2(Ωθ,μ), while the introduction and equations (1)–(2) use H=L2(Ωψ,ν) with a group N acting on ψ=(θ,η). Since Theorem 1's conclusion is the existence of H and the operators Aζ, this ambiguity is material, and the claimed derivation cannot be checked from the manuscript alone.
  4. [Section 2, Proposition 3] The proof that Postulate 5 makes Aθ self-adjoint is incomplete. The Cauchy-Schwarz estimate gives |⟨v|Aθ u⟩|² ≤ I_u I_v, where I_v=∫ |fθ(n)| |⟨v|vn⟩|² dν. For boundedness of the functional v↦⟨v|Aθ u⟩ one needs sup_{∥v∥=1} I_v < ∞ (or an equivalent uniform bound), but the proof only notes that I_v is finite for each fixed v in the domain and then normalizes u and v. Finiteness for each v does not imply uniform boundedness over the unit sphere. The equality D†=D is therefore not established, and the subsequent use of the spectral theorem is not justified.
  5. [Section 2, Theorem 3] Theorem 3 is stated for maximal variables θ and η that are not bijective functions of each other, without a discreteness assumption, but its proof begins "I will prove this for the case of discrete-valued variables" and uses finite or countable sums of rank-one projectors. No argument is supplied for the continuous case, so the theorem as stated is not proved. Since noncommutativity is a central quantum feature, this gap matters for the paper's claims.
  6. [Section 3, Theorem 4] The Born-rule result is asserted rather than proved: the paragraph before Theorem 4 says that "the following version of Born's formula is proved" from Postulates 6 and 7 using a version of Gleason's theorem given by Busch (2003), but no proof appears in the manuscript, no precise statement of the Gleason-type theorem used is given, and the hypotheses are not checked. Theorem 5 is likewise stated without proof. Since the Born rule is one of the paper's advertised consequences and is used in the interpretive sections, this is a substantial omission rather than a local detail.
minor comments (5)
  1. [References, Busch (2003)] The reference to Busch (2003) gives volume 97; the cited "simple proof of Gleason's theorem" appeared in Physical Review Letters 91, 120403.
  2. [References, Jodlicska et al. (2025)] The reference to Jodlicska et al. (2025) contains garbled author names ("Smid, V omlel, J., and Slavik"); the author list should be corrected.
  3. [Postulate 4] The phrase "same category" is glossed as "there is a bijective function connecting Ωθ and Ωη," but in category theory the existence of morphisms in both directions does not generally give a bijection of underlying sets; the intended mathematical meaning should be stated as a definition, not an intuition.
  4. [Introduction and Abstract] The claim that Theorem 1 is "the weakest possible theorem" is not supported by any minimality or lower-bound argument; the wording should be softened or justified.
  5. [Section 2, Lemma 3] The unitarity computation in Lemma 3 is too terse: it appears to assume the adjoint formula rather than deriving it by a change of variables; the step should be spelled out.

Circularity Check

3 steps flagged · score 7.0 of 10

Proposition 1 assumes the inverse it needs to force relatedness; Theorem 1's Hilbert-space conclusion is imported from the author's own Theorem 4 (Helland 2024a), and the Bell/decision theorems are deferred to Helland (2023b).

  1. other [Section 2, Proposition 1 proof]
    "The inverse f −1 is well defined, since, by ξ (φ1) = f (kφ1), and letting φ1 vary, the range of ξ has the same category as the range of η."

    The proof has only assumed that θ = f(φ) for some function f; injectivity of f is not assumed and never established. The same-category condition concerns cardinality or the existence of morphisms between ranges, not containment of one range in another and not invertibility of f. But the proposition's conclusion is precisely that η is a bijective function of the constructed ξ; using f^{-1} to express η(φ1) = f(a(f^{-1}(ξ(φ1)))) smuggles that conclusion into the construction. The definition of ξ as f(kφ) does not by itself make f^{-1} well defined. Thus the reduction of θ and η to a 'related' pair is not proved; it is assumed in the step that is supposed to establish it.

  2. self citation load bearing [Section 2, Theorem 1 proof]
    "Theorem 1 now follows from Theorem 4 of Helland (2024a) and Proposition 2. □"

    The paper's central advertised result — that postulates imply a Hilbert space with a unique symmetric operator for every accessible variable — is not derived in this paper. Its proof ends by invoking Theorem 4 of Helland (2024a), a same-author paper, and the preceding sentence also cites 'Theorem 1 of Helland, 2022a.' The entire existence and uniqueness of the operator assignment is therefore imported from the author's earlier work rather than established from the postulates stated here. Since Proposition 1, the bridge that makes Theorem 4 applicable, is itself unsupported, the load-bearing part of the derivation is a self-citation chain rather than a self-contained proof.

1 more flagged steps
  1. uniqueness imported from authors [Section 5, Theorem 7 proof]
    "Proof: This is a consequence of Theorem 6. See the proof of Theorem 1 in Helland (2023b)."

    The advertised applications — the universal psychological limitation and the explanation of the CHSH/Bell violation — depend on Theorem 7, a uniqueness/limitation theorem whose proof is entirely deferred to another same-author paper, Helland (2023b). The section itself says 'The mathematical proofs are deferred to these articles.' Thus the conclusion that an observer cannot simultaneously entertain a second related variable, and hence the Bell-experiment explanation, is not proved here; it is imported from the author's prior work. This is a load-bearing self-citation chain supporting the central interpretive and empirical consequences of the paper.

full rationale

The derivation chain is not self-contained. Proposition 1's attempted reduction of two same-category maximal variables to a 'related' pair uses f^{-1} although f was only assumed to be a function; it also infers range containment from same category, and it effectively assumes the bijective dependence that the proposition is supposed to establish. Since Theorem 1 opens by invoking exactly that 'related' pair, the foundation is not derived. The core Hilbert-space/operator theorem is then imported from Theorem 4 of Helland (2024a), a same-author paper, with 'Theorem 1 now follows...' — the paper's advertised derivation is a citation, not a self-contained proof; no machine-checked or independently reproduced version is cited. The Bell/decision applications rest on Theorem 7, whose proof is deferred to Helland (2023b). These are load-bearing self-citations, not merely background. There is no fitted-parameter or definitional equivalence, so the circularity is of the self-citation and unsupported-bridge type. Overall score: 7.

Assumptions & free parameters 0 free parameters · 10 assumptions · 3 invented entities

The ledger makes visible what the central derivation assumes without independent support: a universal inaccessible variable, symmetry and category postulates, a convergence postulate for self-adjointness, and several imported same-author theorems. There are no data-fitted free parameters in this paper.

assumptions (10)
  • ad hoc to paper Postulate 1: There exists an inaccessible variable phi such that all accessible variables are functions of phi, with a group M acting on Omega_phi.
    Foundational assumption specific to this reconstruction; no independent evidence is provided.
  • domain assumption Postulate 2: For every accessible variable xi there exists a maximal accessible variable theta with xi <= theta (Zorn's lemma or explicit postulate).
    Assumes the partial order of accessible variables has maximal elements; uses Zorn's lemma.
  • ad hoc to paper Postulate 3: Each accessible theta carries a transitive group G with trivial isotropy and left-invariant measure mu.
    Symmetry structure needed for the representation construction; not a standard physical law.
  • ad hoc to paper Postulate 4: Omega_theta and Omega_eta have the same category (bijective range spaces) for complementary variables.
    Crucial for Proposition 1; imported without physical justification.
  • ad hoc to paper Postulate 5: The integral of |f_theta(n)| times |<u|v_n>|^2 over nu(dn) converges for all |u> in the domain.
    Technical convergence condition assumed to make symmetric operators self-adjoint; not proved from earlier postulates.
  • domain assumption Postulate 6: The generalized likelihood principle holds.
    A statistical principle taken as an axiom; used for the Born rule.
  • ad hoc to paper Postulate 7: A superior actor D with Dutch-book rational probabilities exists, and D's probabilities count as experimental evidence.
    Decision-theoretic construct introduced specifically to derive Born probabilities.
  • ad hoc to paper Postulate 8: Independent events combine by multiplying amplitudes in tensor-product Hilbert spaces.
    Additional rule for composite independent events.
  • ad hoc to paper Theorem 4 of Helland (2024a): existence of a Hilbert space and unique symmetric operators for related maximal variables under a unitary representation condition.
    Imported from a same-author paper; not restated or proved here, but it is the core of Theorem 1.
  • ad hoc to paper Results from Helland (2023b) used in Theorems 6 and 7: unitary equivalence implies relatedness, and relatedness limits an observer's simultaneous variables.
    Same-author results accepted without proof; they carry the Bell and decision-theory consequences.
invented entities (3)
  • inaccessible theoretical variable phi
    purpose: Universal hidden variable from which all accessible variables are functions; carries the group action M.
    Postulate 1 introduces it as a primitive with no direct measurement; its existence is assumed.
  • hypothetical superior actor D
    purpose: Provides rational (Dutch-book) probabilities used to derive the Born rule.
    Postulate 7 introduces D; no empirical handle outside the derivation.
  • theoretical variables (accessible and inaccessible)
    purpose: Primitive notions representing observer knowledge; they are the endpoints of the derivation.
    The paper takes these as primitives with only closure under functions; no independent observational definition is provided.

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Cite this review

Pith. "Pith review of Some mathematical issues regarding a new approach towards quantum foundations." pith.science (2026). https://pith.science/paper/KEWTBANB

@misc{pith2026241113113,
  author       = {Pith},
  title        = {Pith review of: Some mathematical issues regarding a new approach towards quantum foundations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KEWTBANB}},
  note         = {Machine review of arXiv:2411.13113}
}
read the original abstract

In this article, the weakest possible theorem providing a foundation for the Hilbert space formalism of quantum theory is stated. The necessary postulates are formulated, and the mathematics is spelt out in detail. It is argued that, from this approach, a general epistemic interpretation of quantum mechanics is natural. Some applications to the Bell experiment and to decision theory are briefly discussed. The article represents the conclusion of a series of articles and books on quantum foundations.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

7 extracted references · 5 canonical work pages

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    What is a thing?

    Barrett, J. (2007) Information processing in generalized probabilistic theories.Physical Review A 75, 032304. 15 Busch, P. (2003). Quantum states and generalized observables: A simple proof of Gleason’s theorem. Physical Review Letters 97 (12), 120403. Coecke, B. & Paquette, E.O. (2009). Categories for the practicing physicist. arXiv: 0905.3010 [quant-ph]...

    arXiv 2007

  2. [3]

    Helland, I.S. (2024b). On probabilities in quantum mechanics. arXiv: 2401.17717 [quant-ph]. APL Quantum 1, 036116. Helland, I.S. (2024c). A new approach towards quantum foundation and some consequences. 16 arXiv: 2403.09224 [quant-ph]. Academia Quantum

    work page Pith review arXiv 2024

  3. [6]

    Quantum probability for statisticians; some new ideas

    https://doi.org/10.20935/AcadQuant7282. Helland, I.S. (2025). Quantum probability for statisticians; some new ideas. arXiv: 2503.02658 [quant-ph]. Submitted. Helland, I.S. & Parthasarathy, H. (2024). Theoretical Variable, Quantum Theory, Relativistic Quantum Field Theory, and Quantum Gravity. Manakin Press, New Dehli. Jodlicka, P.O., Kos, S., Smid, V omle...

    work page Pith review arXiv 2025

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    Helland, I.S. (2024a). An alternative foundation of quantum mechanics. arXiv: 2305.06727 [quant-ph]. Foundations of Physics 54,

    arXiv 2024

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    Khrennikov, A., Ozawa, M., Benninger, F., and Schor, O. (2025). Coupling quantum-like cog- nition with neuronal netwoorks within generalized probability theory. Journal of Mathematical Psychology 125, 102923. Klein, U. (2010). The statistical origins of quantum mechanics. Hindawi Publishing Corper- ation Physics Physics Research International 2010, 808424...

    work page arXiv 2025

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    Helland; I.S. (2022d). On religious faith, Christianity, and the foundation of quantum mechan- ics. European Journal of Theology and Philosophy 2 (1), 10-17. Helland, I.S. (2023a). On the Foundation of Quantum Theory. The Relevant Articles. Eliva Press, Chisinau, Moldova. Helland, I.S. (2023b). An explanation of the Bell experiment. Journal of Modern and ...

    work page Pith review arXiv 2022

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    Correction (2023) 62,

    work page 2023

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Reviewed August 12, 2026 · model on record in the stance chip above.